In this article, we introduce the notions of sequentially compactness and boundedly compactness in the framework of a newly defined $b_v(s)$-metric space which is a generalization of usual metric spaces and several other abstract spaces. We establish correlations between sequentially compactness and boundedly compactness. Moreover, we prove some fixed point results of contractive mapping in this setting, from which we can deduce several analogous fixed point results. Finally, we illustrate some non-trivial examples to validate the significances and motivations of this manuscript.
Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of {\it saturated} class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed point results. We illustrate the concept by surveying some significant fixed point results obta…
In this paper, we prove the existence of a common end point for a pair of multivalued mappings satisfying a new generalized weakly contractive condition in a complete metric space. Our result generalizes and extends many known results.
We established a fixed-point theorem for mapping satisfying a general contractive inequality of integral type depended an another function. This theorem substantially extend the theorem due to Branciari (2003) and Rhoades (2003)
In this paper, we study the existence of fixed points for mappings defined on complete metric space (X, d) satisfying a general contractive inequality of integral type depended on another function. This conditions is analogous of Banach conditions and Branciari Theorem.
In this short paper, we prove fixed point theorems for nonexpansive mappings whose domains are unbounded subsets of Banach spaces. These theorems are generalizations of Penot's result in [Proc. Amer. Math. Soc., 131 (2003), 2371--2377].
In this paper, we establish some common fixed point results for a new class of pair of contractions mappings having functions as contractive parameters, and satisfying certain commutative properties.
A new, simple and unified approach in the theory of contractive mappings was recently given by Samet \emph{et al.} (Nonlinear Anal. 75, 2012, 2154-2165) by using the concepts of $α$-$ψ$-contractive type mappings and $α$-admissible mappings in metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called generalized $α$-$ψ$ contractive pair of mappings and study various fixed point theorems for such mappings in complete metric spaces. For this, we introduce a new notion of $α$-admissible w.r.t $g$ mapping which in turn generalizes the concept of $g$-m…
In this paper some generalized mappings due to L.B. Ciric over a generalized metric space have been studied with illustrative examples. Some theorems with necessary and sufficient conditions have been proved for such mappings on that space.
The purpose of this paper is to obtain sufficient conditions for the existence of a unique fixed point of T-Kannan type mappings on complete cone metric spaces depended on another function.
In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. \textbf{65} (2006) 1379-1393] and [N.V. Luong and N.X. Thuan, \textit{Coupled fixed points in partially ordered metric spaces and application}, Nonlinear Anal. \textbf{74} (2011) 983-992], by weakening the involved contractive condition. An example as well an application to nonlinear Fredholm integral equations are also given in order to …
In this article, we consider Kannan type contractive self-map $T$ on a metric space $(X,d)$ such that \[d(Tx,Ty)<\frac{1}{2}\{d(x,Tx)+d(y,Ty)\} \mbox{ for all } x \neq y \in X, \] and establish some new fixed point results without taking the compactness of $X$ and also without assuming continuity of $T$. Further, we anticipate a result ensuring the completeness of the space $X$ via FPP of this map. Finally, we are able to give an affirmative answer to the open question posed by J. Górnicki [\textit{Fixed point theorems for Kannan type mappings}, J. Fixed Point Theory Appl. 2017]. Apart from th…
This paper is a continuation to the study of generalized quasi contractive operators, essentially due to Akhtar et al. [A multi-step implicit iterative process for common fixed points of generalized C^{q}-operators in convex metric spaces, Sci. Int., 25(4) (2013), 887-891], in spaces of nonpositive sectional curvature. We aim to establish results concerning convergence characteristics of the classical iterative algorithms such as Picard, Mann, Ishikawa and Xu-Noor iterative algorithms associated with the proposed class of generalized quasi contractive operators. Moreover, we adopt the concept …
The major motives of this paper are to study different types of contractive mappings and also to answer an open question of Garai et al. [The contractive principle for mappings in $b_v(s)$-metric spaces, arXiv:1802.03136]. We first set up some fixed point results associated with two types of contractive mappings in $b_v(s)$-metric spaces and then we give an answer, in positive, to the open question. Most importantly, we characterize the completeness of a $b_v(s)$-metric space via fixed point property of a certain type of contractive mappings. Our results extend and generalized several importan…
We introduce multivalued Górnicki mapping and various new types of multivalued enriched contractive mappings, viz., multivalued enriched Kannan mapping, multivalued enriched Chatterjea mapping, and multivalued enriched Cirić-Reich-Rus mapping. Then we establish some existence results for the fixed points of these multivalued contractive type mappings using the fixed point property of the average operator of the mappings. Furthermore, we present some data dependence results for the fixed points of mentioned classes of mappings.
We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalised convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically $ψ$-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalise and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitativ…
We propose a general approach to defining a contractive-like multivalued mappings $F$ which avoids any use of the Hausdorff distance between the sets $F(x)$ and $F(y)$. Various fixed point theorems are proved under a two-parameter control of the distance function $d_{F}(x)=dist(x,F(x))$ between a point $x \in X$ and the value $F(x) ßX$. Here, both parameters are numerical functions. The first one $\a\,:[0,+ı)\rightarrow [1,+ı)$ controls the distance between $x$ and some appropriate point $y \in F(x)$ in comparison with $d_{F}(x)$, whereas the second one $\b\,:[0,+ı)\rightarrow [0,1)$ estimates…
The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for p in (0, 1]. More precisely, based on the fixed point theorem of single-valued continuous condensing mapping in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) se…
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of {\it enriched nonexpansive mappings} in Hilbert spaces. In order to approximate the fixed points of enriched nonexpansive mappings we use the Krasnoselskij iteration for which we prove strong and weak convergence theorems. Examples to illustrate the richness of the new class of contractive mappings are also given. Our results in this paper extend some classical convergence theorems established by Browder and Petryshyn in [Browder, F. …
For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipsch…
In 2007 H. Long-Guang and Z. Xian, [H. Long-Guang and Z. Xian, Cone Metric Spaces and Fixed Point Theorems of Contractive Mapping, J. Math. Anal. Appl., 322(2007), 1468-1476], generalized the concept of a metric space, by introducing cone metric spaces, and obtained some fixed point theorems for mappings satisfying certain contractive conditions. The main question was "Are cone metric spaces a real generalization of metric spaces?" Throughout this paper we answer the question in the negative, proving that every cone metric space is metrizable and the equivalent metric satisfies the same contra…
In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on n- normed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an n-normed space.
In this paper, we study the existence of fixed points for mappings defined on complete, (sequentially compact) cone metric spaces, satisfying a general contractive inequality depending of two additional mappings.
In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric spaces are Hausdorff and also first countable. Moreover, we assert that every separable $\mathcal{F}$-metric space is second countable. Additionally, we acquire some interesting fixed point results concerning altering distance functions for contractive-type mappings and Kannan-type contractive mappings in this exciting context. However, most of the findings are well-furnished by several non-trivial numerical examples. Finally, we raise an open problem regarding the metrizability of such kind of spaces.
Fixed Point Theorems for Set-Valued Maps with Contractive Orbits
Faculty of Mathematics and Informatics, Sofia University（索非亚大学数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息研究所）
Our aim in this paper is to present a new type of the modular space. This space contains the classical modular space. There are some mappings that do not have contractive condition in the usual modular space but become contraction in this new space.
We introduce a large class of contractive mappings, called Suzuki Berinde type contraction. We show that any Suzuki Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mapping is also obtained. These results unify, generalize and complement various known comparable results in the literature.
In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. TMA \textbf{65} (2006) 1379-1393] by significantly weakening the involved contractive condition. Our technique of proof is essentially different and more natural. An example as well an application to periodic BVP are also given in order to illustrate the effectiveness of our generalizations.
In this paper, we establish coincidence fixed point and common fixed point theorems for two mapping in complete $C^*$-algebra-valued metric spaces which satisfy new contractive conditions. Some applications of our obtained results are given.
A contractive condition is addressed for extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. It is proven that if the space is uniformly convex and the subsets are non-empty, closed and convex then all the iterations converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduce to a unique fixed point if all such subsets intersect.
The main goal of this note is to show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants. As an application, a contractive multiplication operator is either a multiplication with a constant, or is completely non-unitary. Additionally we explore possibilities for a multiplication operator to be (weakly) compact and (uniformly) mean ergodic.
We introduce a large class of mappings, called enriched contractions, which includes, amongst many other contractive type mappings, the Picard-Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Kransnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results in this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia type…
In this article, we investigate for some sufficient conditions for the existence and uniqueness of best proximity points for the topological $p$-proximal contractions and $p$-proximal contractive mappings on arbitrary topological spaces. Moreover, our results are authenticated by a few numerical examples and these generalize some of the known results in the literature.
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, introduced in [Berinde, V., {\it Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces}, Carpathian J. Math. {\bf 35} (2019), no. 3, 277-288.], we introduce the class of enriched Chatterjea contractions and prove general fixed point theorems for such contractions in the setting of a Banach space. Examples to illustrate the richness of the new class of contractions and the relationship between enriched Banach contractions, enriched Kannan contractions an…
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
In this paper, the notion of $\mathbb{C}$-simulation function is introduced and the existence and uniqueness of common fixed points of two self-mappings satisfying contractive conditions in the setting of complex valued metric spaces via $\mathbb{C}$-simulation functions are studied. Examples are also provided to demonstrate the results. The existence and uniqueness of a first-order periodic differential equation is also obtained as an application of the result.
In this paper the concept of a partial cone metric space is investigated, some continuity type theorems, and fixed point theorems of contractive mappings in this generalized setting are proved as well as some theorems related to topological properties.
We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space $X$ into $X$, and the second with an application of the continuation method to the case where they satisfy the Leray-Schauder boundary condition in Banach spaces.
This paper discusses a more general contractive condition for a class of extended cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are non-empty, closed and convex then all the iterates converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduces to a unique fixed point if all such subsets intersect.
In this paper, we prove some common coupled fixed point theorems for mappings satisfying different contractive conditions in the context of complete $C^*$-algebra-valued metric spaces. Moreover, the paper provides an application to prove the existence and uniqueness of a solution for Fredholm nonlinear integral equations.
We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $β$, and then give convergence theorems for martingales of $β$-conditional expectations. We give the Birkhoff ergodic theorem for $β$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $β$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of…
